A Reformulation of the Hoop Conjecture

A reformulation of the Hoop Conjecture which avoids the use of any foliations as well as ill-defined notions of “encircled mass” and “circumference of a body” is presented. The new formulation is based on the concept of trapped circle, which is well-defined, geometrically sound, and physically relevant. With the new formulation, the problems of severe compactness in every spatial direction, and of how to superpose the hoops with the surface of the black hole, are resolved. A new conjecture concerning “peeling” properties of dynamical/trapping horizons that may form in realistic gravitational collapse is put forward. Finally, I discuss the possibility of carrying over the results to arbitrary dimension and comment on how to deal with non-spherical topologies, in particular with black rings. PACS: 04.70.Bw, 04.20.Dw, 04.20.Cv, 04.50.+h Among the various —yet unproven— beliefs concerning the physics of black holes, a very popular one is the Hoop Conjecture formulated, in a deliberately vague manner, as [66, 49]: “black holes with horizons form when, and only when, a mass M gets compacted into a region whose circumference in every direction is C . 4πGM/c.” The impreciseness of this statement was already remarked at its birth [66, 49], and has been discussed many times, e.g. [57, 26, 68], for many physical and mathematical difficulties arise. The main problems can be summarized as: i. in practice, it is almost impossible to determine the existence of an event horizon, which is a global concept only to the avail of omniscient observers. The remedy in this case has been the use of alternative, local, definitions of horizons, mainly apparent horizons [32, 43], based on the concept of closed trapped surface first introduced in [53]. Nowadays, the newer notions of dynamical/trapping horizons [3, 34] seem to be a reasonable alternative. One therefore deals with the simpler and more fundamental notion of (a world tube of) closed marginally trapped surfaces. I will follow this strategy here.